Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)

Technical Summary: Hochschild Cohomology of Beilinson Algebras of Graded Down-Up Algebras with Weights (n, m)

Problem Statement
This paper addresses the computation of the Hochschild cohomology groups, $HH^r(\nabla A)$, for the Beilinson algebra $\nabla A$ associated with a graded down-up algebra $A = A(\alpha, \beta)$. The algebra $A$ is defined over an algebraically closed field $k$ of characteristic zero with generators $x, y$ and weights $(\deg x, \deg y) = (n, m)$. The study focuses on the case where $\beta \neq 0$, ensuring $A$ is a 3-dimensional noetherian AS-regular algebra, and specifically targets the remaining unresolved case where the weights satisfy $m \ge n > 1$ and $\gcd(n, m) = 1$.

Previous work had established results for the cases $(n, m) = (1, 1)$ (Belmans) and $(n=1, m \ge 2)$ (Itaba–Ueyama). The current work extends these findings to the scenario where both weights are greater than 1. The primary objectives are to determine the dimensions of the Hochschild cohomology groups, construct explicit bases for $HH^1$ and $HH^2$, describe the ring structure with respect to the Yoneda product, and investigate the derived category equivalence of the associated noncommutative projective scheme.

Methodology
The authors employ an algebraic approach utilizing the minimal projective resolution of the Beilinson algebra $\nabla A$ as a module over its enveloping algebra $(\nabla A)^e$. The methodology proceeds through the following steps:

1. Resolution Construction: Following the framework of Green–Snashall, the authors construct the minimal projective resolution $P_\bullet$ of $\nabla A$. They define specific sets of generators $G_0, G_1, G_2$ corresponding to vertices, arrows, and relations of the quiver of $\nabla A$. The differentials $\partial_1$ and $\partial_2$ are explicitly defined.
2. Dual Complex and Matrix Representation: By applying the functor $\text{Hom}_{(\nabla A)^e}(-, \nabla A)$, the authors obtain a complex of dual spaces. They construct $k$-bases for these dual spaces and represent the induced maps $\widehat{\partial}_1$ and $\widehat{\partial}_2$ as matrices ($L_1$ and $L_2$).
3. Rank Computation: The dimensions of the cohomology groups are derived by computing the ranks of these representation matrices. The authors utilize properties of circulant matrices and their associated polynomials to determine the ranks of $L_1$ and $L_2$ under various conditions on the parameters $\alpha, \beta$ and the weights $n, m$.
4. Basis Construction: Explicit bases for the cohomology groups $HH^1(\nabla A)$ and $HH^2(\nabla A)$ are constructed by analyzing the kernels and images of the differentials in the dual complex.
5. Ring Structure: The Yoneda product structure on the total Hochschild cohomology ring $\bigoplus_{r \ge 0} HH^r(\nabla A)$ is determined by constructing chain maps that lift the cohomology classes and computing their compositions.
6. Derived Category Analysis: The authors utilize the Euler characteristic of the Hochschild cohomology and the properties of the Serre functor on the Grothendieck group to test for unipotency, comparing the results with known theorems regarding derived categories of smooth projective varieties.

Key Results

• Dimension Formulas: The paper provides a complete dimension formula for $HH^r(\nabla A)$ for all $r \ge 0$ in the case $m > n > 1$ with $\gcd(n, m) = 1$.
  • $HH^0(\nabla A) \cong k$ (dimension 1).
  • $HH^1(\nabla A)$ has dimension 2 if $n+m$ is even and $\alpha=0$ (Case I), and dimension 1 otherwise (Case II).
  • $HH^2(\nabla A)$ dimensions depend on $n$ and the case:
    • If $n=2$: dimension is $m+5$ (Case I) or $m+4$ (Case II).
    • If $n \ge 3$: dimension is $n+m+1$ (Case I) or $n+m$ (Case II).
  • $HH^r(\nabla A) = 0$ for all $r \ge 3$.
• Explicit Bases: The authors construct explicit $k$-bases for $HH^1(\nabla A)$ and $HH^2(\nabla A)$ for all sub-cases (including specific conditions on $\alpha$ and $\beta$ such as $\lambda_{m+1}=0$ or $\alpha^2+4\beta=0$).
• Ring Structure: The total Hochschild cohomology ring is shown to be isomorphic to a quotient of an exterior algebra $\Lambda(a, b)$ by a homogeneous ideal $I$. The number of generators $(a, b)$ and the specific relations in $I$ are tabulated for all cases. For many cases (particularly when $n \ge 2$), the ideal $I$ is trivial, implying the ring is a free exterior algebra.
• Derived Category Non-Equivalence: A significant structural consequence is derived for the case $m > n > 1$. The authors calculate the Euler characteristic $\chi_{HH}(HH^\bullet(\nabla A))$ and show it does not equal the rank of the Grothendieck group of the bounded derived category of the noncommutative projective scheme, $2(n+m)$. By invoking a theorem of Bondal–Polishchuk regarding the unipotency of the Serre functor on smooth projective surfaces, they conclude that the Serre functor does not act unipotently. Consequently, the derived category $Db(\text{tails } A)$ is not equivalent to the bounded derived category of any smooth projective surface.

Significance and Claims
The paper claims to extend the classification of Hochschild cohomology for Beilinson algebras of graded down-up algebras to the previously unaddressed case where both weights are greater than 1. By explicitly constructing projective resolutions and computing matrix ranks, the authors provide a complete algebraic description of the cohomology dimensions and ring structures.

The work highlights a distinction in the geometric nature of these noncommutative algebras. While the cases $(1,1)$ and $(1,2)$ (where $n=1$) exhibit derived equivalences to smooth projective surfaces (or have unipotent Serre functors), the paper demonstrates that for $m > n > 1$, this equivalence breaks down. This result serves as a counterexample to the universality of derived equivalences between AS-regular algebras and smooth projective varieties in higher-weight scenarios, refining the understanding of the geometric realization of noncommutative projective schemes.